Finite Group Databasebeta

← #10863706

Subgroups — #10863706

Order 768.

804 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83173C211yesmaximal, characteristic
2#76039C211yesmaximal, characteristic, complemented
3C3×#2041C211yesmaximal, characteristic
4#71762C211yesmaximal, characteristic
5#70186C211yesmaximal, characteristic, complemented
6#70180C211yesmaximal, characteristic, complemented
7#70169C211yesmaximal, characteristic, complemented
8#70169C211yesmaximal, characteristic, complemented
9#70166C211yesmaximal, characteristic, complemented
10#70155C211yesmaximal, characteristic, complemented
11#70154C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69601C211yesmaximal, characteristic, complemented
14#68624C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#26753—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#389C2211yescharacteristic
26#5221—21no—
27#5221C2211yescharacteristic
28#5221C2211yescharacteristic
29#5220—21no—
30#5220—21no—
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×C8×Dic3C2211yescharacteristic
38#4981—21no—
39#4980—21no—
40#4977C2211yescharacteristic
41Dic3×C4⋊C4C2211yescharacteristic
42#4972C2211yescharacteristic
43#4972C2211yescharacteristic
44C2×#808C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47#4928C2211yescharacteristic
48#4658—21no—
49#4650—21no—
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2508—31no—
61#2366—31no—
62#2073—31no—
63#2067—31no—
64#2041—31no—
65#2041D311yescharacteristic
66#1934—31no—
67#1925—31no—
68#1921—31no—
69#1919—31no—
70#1916—31no—
71#1864—31no—
72#1864—31no—
73#1804—31no—
74C4×#102—31no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#127—22no—
84C3×#127—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95D4×Dic3—22no—
96D4×Dic3—22no—
97D4×Dic3—41no—
98D4×Dic3—41no—
99D4×Dic3—44no—
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3D411yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C3×#102—22no—
116C3×#102—22no—
117C3×#102—41no—
118C3×#102—41no—
119C3×#95—21no—
120#832—22no—
121#832—22no—
122#832—22no—
123#832—22no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#818—22no—
134Dic3⋊C8—41no—
135C8×Dic3—41no—
136#808—22no—
137C2×#127C2×D311yescharacteristic
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4—61no—
141C2×#107—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×#102C2×D311yescharacteristic
151C2×#102C2×D311yescharacteristic
152C2×C4×C8—31no—
153#400—61no—
154#398—61no—
155#390—31no—
156#390—31no—
157#390—61no—
158#389—31no—
159#389—31no—
160#389—31no—
161#389C2×D311yescharacteristic
162#387—61no—
163#386—61no—
164#386—61no—
165#382—31no—
166#382—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
180C24×C3—21no—
181C24×C3—21no—
182C2×C3×D4—24no—
183C2×C3×D4—24no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—42no—
195C2×C3×D4—42no—
196C2×C3×D4—81no—
197C2×C3×D4—81no—
198C2×C3×D4#4112yes—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—81no—
214C22×Dic3—81no—
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—42no—
230C3×C42—21no—
231C3×C42—21no—
232#217—42no—
233#217—42no—
234#217—42no—
235#217—42no—
236#217—81no—
237#217—81no—
238C4⋊Dic3—22no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—22no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3#4112yes—
247C4⋊Dic3#4112yes—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C22×D4—21no—
264C22×D4—21no—
265C22×D4C22×D311yescharacteristic
266C22×D4C22×D311yescharacteristic
267C23×C4—61no—
268C23×C4—61no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8C22×D311yescharacteristic
272C22×C8C22×D311yescharacteristic
273#127—22no—
274#127—22no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—121no—
278C4×D4—121no—
279C4×D4—124no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C42—31no—
299C2×C42—31no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—62no—
304#106—62no—
305C4⋊C8—121no—
306#102—22no—
307#102—22no—
308#102—41no—
309#102—41no—
310#102—62no—
311#102—62no—
312#102—62no—
313#102—62no—
314C4×C8—121no—
315#95—21no—
316#95—31no—
317#95—31no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C22×D411yescharacteristic
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—22no—
343C3×D4—22no—
344C3×D4—48no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—84no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4#13512yes—
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367C2×C3×C42−1+411yescharacteristic
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—84no—
392C2×Dic3—84no—
393C2×Dic3AGL(1,ℤ/8)12yes—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398C24—21no—
399C24—21no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—81no—
415C2×D4—81no—
416C2×D4#23712yes—
417C2×D4#23712yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—241no—
433C22×C4—241no—
434C22×C4D3×D411yescharacteristic
435C22×C4D3×D411yescharacteristic
436C22×C4C23×D311yescharacteristic
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—62no—
457C4⋊C4—62no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—241no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C22⋊C4—122no—
468C22⋊C4—122no—
469C22⋊C4—122no—
470C22⋊C4—122no—
471C22⋊C4—241no—
472C22⋊C4—241no—
473C42—21no—
474C42—21no—
475C42—61no—
476C42—61no—
477C42—61no—
478C42—61no—
479C42—122no—
480C42—122no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3#52511yescharacteristic
496C22×C3#54611yescharacteristic
497C22×C3#54811yescharacteristic
498C22×C3C2×#13511yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57611yescharacteristic
501C22×C3#57711yescharacteristic
502C3×C4—22no—
503C3×C4—41no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4#52911yescharacteristic
507C3×C4#53911yescharacteristic
508C3×C4#55211yescharacteristic
509C3×C4#55311yescharacteristic
510Dic3—22no—
511Dic3—42no—
512Dic3—41no—
513Dic3—41no—
514Dic3—41no—
515Dic3—42no—
516Dic3—81no—
517Dic3—81no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533D4—22no—
534D4—22no—
535D4—22no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—84no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—32no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—62no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—244no—
579C2×C4—244no—
580C2×C4#91212yes—
581C2×C4C2×#23711yescharacteristic
582C2×C4C2×#23711yescharacteristic
583C2×C4#100411yescharacteristic
584C2×C4D3×#4111yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4#101011yescharacteristic
587C8—41no—
588C8—41no—
589C8—121no—
590C8—121no—
591C2×C3—82no—
592C2×C3—82no—
593C2×C3#246011yescharacteristic
594C2×C3#310511yescharacteristic
595C2×C3#311811yescharacteristic
596C2×C3#330511yescharacteristic
597C2×C3#331011yescharacteristic
598C2×C3#336511yescharacteristic
599C2×C3#336611yescharacteristic
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22#558611yescharacteristic
615C22#561211yescharacteristic
616C22#567711yescharacteristic
617C22C2×#91211yescharacteristic
618C22#576211yescharacteristic
619C22D3×AGL(1,ℤ/8)11yescharacteristic
620C22#577411yescharacteristic
621C4—22no—
622C4—41no—
623C4—62no—
624C4—81no—
625C4—81no—
626C4—121no—
627C4—122no—
628C4—122no—
629C4—121no—
630C4—121no—
631C4—241no—
632C4—241no—
633C4#560111yescharacteristic
634C4#560211yescharacteristic
635C4#564911yescharacteristic
636C4#565011yescharacteristic
637C3#2675311yescharacteristic, complemented, Sylow
638C2—82no—
639C2—82no—
640C2#7691311yescharacteristic
641C2#7998911yescharacteristic
642C2#8013611yescharacteristic
643C2#8035311yescharacteristic
644C2#8039711yescharacteristic
645C2#8158911yescharacteristic
646C2#8170811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2041D3
hypocenterC3#26753
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #76039C2semidirect
5. #70186C2semidirect
6. #70180C2semidirect
7. #70169C2semidirect
8. #70169C2semidirect
9. #70166C2semidirect
10. #70155C2semidirect
11. #70154C2semidirect
12. Dic3×#102C2semidirect
13. #69601C2semidirect
14. #68624C2semidirect
15. #68610C2semidirect
637. C3#26753semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#831732
#760392
C3×#20412
#717622
#701862
#701802
#701692
#701662
#701552
#701542
Dic3×#1022
#696012
#686242
#686102
#267533